If equation of tangent at  $P,Q$  and vertex  A  of a parabola are  $3x+4y-7=0,2x+3y-10=0$  and   $x-y=0$ respectively, then

Parabola Tangent Problem

Given Information:

• Tangent at P: 3x + 4y - 7 = 0
• Tangent at Q: 2x + 3y - 10 = 0
• Tangent at vertex A (axis): x - y = 0

Analysis:

For a parabola, all tangents are related through its geometric properties. The axis of the parabola is given as the tangent at vertex A.

Axis of parabola: x - y = 0

This means the parabola opens along the line y = x.

Finding the Vertex:

The vertex lies on the axis, so it satisfies x - y = 0, which means x = y.

The vertex can be found by solving the system of the axis equation with one of the tangents:

x - y = 0 ⟹ x = y

Substitute into the tangent at P:

3x + 4y - 7 = 0 
3x + 4x - 7 = 0 (since x = y) 
7x = 7 
x = 1, y = 1

Therefore, the vertex is at (1, 1).

Properties of the Parabola:

• Axis: x - y = 0 or y = x
• Vertex: (1, 1)
• Direction: Opens along the axis y = x

Verification:

We can verify that the vertex (1, 1) lies on the axis:

x - y = 1 - 1 = 0 ✓

Key Results:

• Axis of parabola:x - y = 0
• Vertex:(1, 1)
• Parabola opens:Along the line y = x

Note: Further properties like focus, directrix, and complete equation can be determined using additional geometric relationships if needed.