The questioner wanted "a geometric way" solution instead of eyeballing. Unfortunately the problem is loosely defined.See the next image:
The question contains 2 circles on the same Y-coordinate - the purple one and the red one in my illustration. Then there's a horizontal line (blue) from the top point C of the red circle to the purple circle point X.
The questioner seems to want a curve AB (orange) between the purple circle and the line CX so that the curve is tangential with the purple circle in A and with the line in point B somewhere between C and X. The places of A and B were not defined and the exact wanted type of curve AB is also undefined.
There's already an answer which suggests rounding corner X with the corner widget. Tangency points A and B will take their places as the widget wants and depending on what's the rounding radius. Actually there's only one possible circular arc and point pair AB if the radius of the arc is given beforehand. In legacy Illustrator and freeware the arc can be constructed exactly also without the corner widget. One can use elementary geometry, but that's tricky when compared to the corner widget.
Another approach is to draw manually a 2 node Bezier curve with the pen. The tangency is not a problem. One can turn the handles on the tangent lines. The tangent line in point A can be got by rotating 90 degrees a radius of the purple circle. Points A and B can be selected freely because AB will be a Bezier curve, not an arc:
Point A was selected arbitarily. A new anchor was inserted to it on the circle. The yellow radius and its 90 degrees rotated copy (green) were drawn to A. The yellow curve was drawn with the pen by dragging the handles along the green line and line XC (=cyan to allow the handles to be seen). The handles snap during the drawing if one has Smart Guides and Snap to Point ON, no other snaps!
I'm afraid this is not what's wanted. The tangency is achieved, but the result has too much variation possibilities which essentially is the same as the questioner has already tried.
The 3rd idea is to declare that AB is a circular arc, but its endpoint B is decided beforehand. It can be anywhere between X and C, even the same as point C. The next image is used to describe the construction:
The horizontal distance between point B and the center of the purple circle is declared to be = P.
If P and the radiuses R1 and R2 are known as numbers one can calculate the radius of the arc (=R3) from the Pythagorean triangle equation. Then it can be used with the corner widget or as well to draw the wanted arc (actually the full circle) as a plane geometry construction.
The rectangular triangle to be used is the cyan one in the next image:
The Pythagorean theorem binds together P and the radiuses. The starting equation and the solved arc radius R3 are shown in the next image:
We can try it for ex. to draw the case where R1 = 30 millimeters, R2 = 20 millimeters and P = 40 millimeters. Substituting R1, R2 and P to the formula gives R3 = 55 millimeters. It's an integer. If there were decimals, one must NOT truncate them. We draw circles with diameters 60, 40 and 110 millimeters. To place the 110 mm circle (green) we insert a high enough 40 mm (=P) wide rectangle one corner in the center of the 30 mm circle. The center of the green circle snaps to the right vertical side of the rectangle and its bottom node snaps to the line XC:
One can stretch the rectangle downwards, draw the copies of the circle and the line XC also to the bottom side and collect the final shape with the Shape Builder tool. The result:
NOTE: The placement and the size of the green circle does not depend on how far to the right the red circle was placed from the center of the purple circle. As said point B can well be the same as C. Here's a small horizontal zone left between B and C.
The 4th idea is to show that my former clause "freeware doesn't have a corner widget" was false. A free CAD program DesignSpark Mechanical has it. It has recently got also constraint based sketching which also could be used, but the corner rounding tool is enough for this job.
Here's a horizontal line and 2 circles on it. The grid snapping is not used, so the items are not aligned with the grid. But the drawn parts snap to each other. The horizontal lines are inserted to the top and bottom of the smaller circle. The vertical line is the wanted ending of the forthcoming rounding.
The rounding tool is applied. One clicks to the big circle and the wanted ending point of the rounding. The old lines vanish automatically; they are rounded:
One can wipe off the unnecessary splinters with the trim tool:
The program is freeware. It has limitations which can be removed by paying. One of them is exporting drawings. But as a workaround one can print the scene as PDF and open the image as vector in Illustrator for color and line type selection.
BTW. Although limited, the program still is a CAD program with very useful 3D capabilities. I would try to draw there as much as possible of the job.