Let me extract and solve each part of the question from the uploaded image. Here's what the image includes:

Problems

1. Solve for the equation $x^2 + y^2 - 7x + 5y - 14 = 0$, passing through $(5, 3)$.
2. Confirm if $(-7, -2)$ lies on the circle $(x+1)^2 + (y-2)^2 = 26$.
3. Find the tangential distance from a point to the given circle.
4. Solve for the equation of a parabola: $x^2 + y^2 - 4x - 12 = 0$.
5. Find the equation of the line $y = z$, passing through points $(0, 4)$ and $(3, -2)$.

Step-by-step Solutions:

1. Solve $x^2 + y^2 - 7x + 5y - 14 = 0$, given point $(5, 3)$:

Substitute $(x, y) = (5, 3)$ into the equation to check if the point satisfies it: $x^2 + y^2 - 7x + 5y - 14 = 5^2 + 3^2 - 7(5) + 5(3) - 14$ $= 25 + 9 - 35 + 15 - 14 = 0$ Since the result is 0, $(5, 3)$ lies on the curve.

2. Check if $(-7, -2)$ lies on the circle $(x+1)^2 + (y-2)^2 = 26$:

Substitute $(x, y) = (-7, -2)$ into the circle equation: $(x+1)^2 + (y-2)^2 = (-7+1)^2 + (-2-2)^2 = (-6)^2 + (-4)^2 = 36 + 16 = 52$ The result is 52, not 26. Hence, $(-7, -2)$ does not lie on the circle.

3. Find tangential distance from a point to the circle:

Let the circle be $(x+1)^2 + (y-2)^2 = 26$, with center $(-1, 2)$ and radius $\sqrt{26}$. The tangential distance from a point $(x_1, y_1)$ to the circle is: $\text{Distance} = \left| \sqrt{(x_1 - h)^2 + (y_1 - k)^2} - r \right|$ Where $(h, k)$ is the center of the circle. Let me know the point from which the tangential distance is needed.

4. Solve for the parabola $x^2 + y^2 - 4x - 12 = 0$:

Rewrite in standard form: $x^2 - 4x + y^2 = 12$ Complete the square for $x$: $(x^2 - 4x + 4) + y^2 = 12 + 4 \quad \implies \quad (x-2)^2 + y^2 = 16$ This is a circle with center $(2, 0)$ and radius $4$, not a parabola.

5. Equation of the line passing through $(0, 4)$ and $(3, -2)$:

The slope $m$ of the line is: $m = \frac{-2 - 4}{3 - 0} = \frac{-6}{3} = -2$ Using the point-slope form of the equation $y - y_1 = m(x - x_1)$, with $(x_1, y_1) = (0, 4)$: $y - 4 = -2(x - 0) \quad \implies \quad y = -2x + 4$ Thus, the line's equation is $y = -2x + 4$.

Let me know if you'd like any clarifications or further details on these solutions!

Follow-up Questions:

1. How do you find the center and radius of a circle from its general equation?
2. What is the formula for finding the tangential distance from a point to a circle?
3. How do you confirm if a point lies on a given curve or circle?
4. What is the method for completing the square for conic sections?
5. How do you derive the equation of a line from two points?

Tip:

Always verify your solutions by substituting back into the original equations to ensure correctness.